Abstract

Let G be a group. We say that an element f ∈ G is reversible in G if it is conjugate to its inverse, i.e. there exists g ∈ G such that g−1 fg = f−1. We denote the set of reversible elements by R(G). For f ∈ G, we denote by Rf(G)the set (possibly empty) of reversers of f, i.e. the set of g ∈ G such that g−1fg = f−1. We characterise the elements of R(G) and describe each Rf(G), where G is the the group of biholomorphic germs in one complex variable. That is, we determine all solutions to the equation f o g o f = g, in which f and g are holomorphic functions on some neighbourhood of the origin, with f(0) = g(0) = 0 and f'(0) ≠ 0 6 ≠ g' (0).